List of top Mathematics Questions

Find the angle between the vectors \[ \vec{a}=\hat{i}+\hat{j}-\hat{k} \] and \[ \vec{b}=\hat{i}-\hat{j}+\hat{k}. \]

Consider an infinite series with first term a and common ratio r. If its sum is 4 and the second term is \(\frac{3}{4}\), then

The substitution required to reduce the differential equation \( \frac{dy}{dx}+\sin y \cos y \sin x = \sin 2x \cos^{2}y \) to a linear differential equation in z is

The differential equation among the following, whose general solution is \( y=A e^{5x}+B e^{-4x} \) is

If \( \int_{0}^{3} x\left(\sqrt[5]{9-x^{2}}\right) \, dx = k \cdot 3^{1/k} \), then \( k = \)

If \( \int \frac{2x+5}{(x-1)(x+1)(x+4)(x+6)} \, dx = \frac{1}{10}\log\left(\frac{f(x)}{g(x)}\right)+c \) and \( \frac{f(-2)}{g(-2)}=6 \), then \( \frac{f(10)}{g(10)} = \)

\( \lim_{n \rightarrow \infty} \frac{1}{n}\sum_{r=1}^{n}\sin^{k}\left(\frac{\pi r}{2n}\right)\cos\left(\frac{\pi r}{2n}\right) = \)

The general solution of the differential equation \( \frac{dy}{dx}=x^{2}y(x+y+xy+1) \) is \( y = \)

If \( \int_{\pi/4}^{3\pi/4} \frac{x \sin x}{1+3\cos 2x} \, dx = k \int_{\pi/4}^{3\pi/4} \frac{\sin x}{1+3\cos 2x} \, dx \), then \( \int_{0}^{k} \sin^{\pi/k}x \, dx = \)

The area (in sq. units) of the region bounded between the curve \( y=x^{2}+5x+1 \) and the line \( 7x-y+1=0 \) is

Approximate value of \( \sqrt[3]{345} \), when it is calculated with the application of derivatives, is

The length of the normal drawn to the curve \( 2x^{3}+2y^{3}=9xy \) at the point (2, 1) is

If \( \int e^{5x}x^{n} \, dx = F(n, x) + c \), then \( 5F(n, x) + nF(n-1, x) = \)

If \( F(x)=\int x(\log x)^{2} \, dx \) and \( F(e)=\frac{e^{2}}{4} \), then \( F(1) = \)

If \( \int e^{x}\left(\frac{1}{n}+\tan nx\right)\sec nx \, dx = \frac{1}{n}(g(x)+k) = F(x) \) and \( F(0)=1 \), then \( k = \)

For \( x>0 \), if \( \int \frac{1}{x^{2}+5x+7} \, dx = \frac{2}{\sqrt{3}}F(x)+k \) and \( F\left(-\frac{5}{2}\right)=0 \), then \( \sin(F(x)) = \)

If \( x^{2}y - xy^{2} + x^{3} - y^{3} = 0 \), then \( \frac{dy}{dx} \) at the point (1, 1) is

If \( y=\text{sech}^{-1}\left(\frac{9}{9x^{2}+10}\right) \), then \( \frac{dy}{dx} = \)

If the rate of increase in the surface area of a cube is 6 sq. cm./sec., then the rate of increase in its volume (in c. c./sec), when the length of its edge is 12 cm, is

If a sector of maximum area is made with a wire of length 40 cm, then the area (in sq cms) of that sector is

If \( f(x)=|x-2|(3^{4|x|}-1) \) is a real valued function, then the set of points at which f is not differentiable, is

If the function \[ f(x)=\begin{cases}\frac{e^{b(x-1)^{2}}-1}{\sqrt{x^{2}-1}} & , \text{for } x>1 \\ \sqrt{2} & , \text{for } x=1 \\ \log\left(\frac{1+bx}{1-bx}\right)\frac{1}{\sin^{2}x} & , \text{for } 0<x<1 \end{cases} \] is continuous at \( x=1 \), then \( \lim_{x \rightarrow 2} \frac{x^{2}-5x+6}{x-2} = \)

\( A(1,2,3) \), \( B(3,4,k) \), \( C(2,1,4) \) form an isosceles triangle. If \( AB=BC \), then the area of \( \triangle ABC \) is

The value of \( \lim_{x\rightarrow0}\frac{e^{2x^{2}}-\cos 2x}{x^{2}} = \)

If the angle between the planes \( \lambda x-2y+3z+1=0 \) and \( 2x+3y-\lambda z+\lambda=0 \) is \( \cos^{-1}(\frac{12}{49}) \) and \( \lambda\in\mathbb{Z} \), then the sum of the perpendicular distances from the origin to these planes is